System and method for measuring second order and higher gradients

ABSTRACT

A system and method of effectively measuring a change in a gradient of a magnetic field. The systems include a first magnet and a second magnet mechanically coupled together and aligned along a polarization axis. The first magnet and the second magnet are positioned such that a pair of like magnetic poles of the first magnet and the second magnet face in opposite directions. Further, the first magnet and the second magnet are configured to move along the polarization axis in response to a magnetic field. A sensing system is configured to measure a change in a gradient of the magnetic field based on the movement of the first magnet and second magnet along the polarization axis in response to the magnetic field.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority to and the benefit of U.S. ProvisionalPatent Application No. 63/159,829, filed on Mar. 11, 2021, and entitled“CASIMIR-ENABLED QUANTUM MAGNETOMETER”, and U.S. Provisional PatentApplication No. 63/300,858, filed Jan. 19, 2022, and entitled “HIGHERORDER GRADIOMETERS AND USES THEREOF”, and U.S. Provisional PatentApplication No. 63/300,907, filed on Jan. 19, 2022, and entitled “FIRSTORDER SINGLE-POINT GRADIOMETER (FOG) AND USES THEREOF”, the contents ofwhich are incorporated herein by reference as though fully set forthherein.

GOVERNMENT SUPPORT

This invention was supported, in whole or in part, with United Statesgovernment support under Grant No. 1647837 awarded by the NationalScience Foundation. The United States Government has certain rights inthis invention.

FIELD OF THE TECHNOLOGY

The subject disclosure relates to sensing systems configured to measureforces uses gradiometers of the second order or greater.

BACKGROUND OF THE TECHNOLOGY

There is a need for more accurate measurements of magnetic fields. Mostexisting magnetic sensors detect field intensity. Existing sensorsinclude SQUID sensors, atomic magnetometers, and magnetoresistivesensors. These existing sensors that rely on magnetic field intensitiessuffer from various inaccuracies, as will be discussed in more detailbelow.

Existing magnetic arrays comprise the arrangement of permanent magnetsto create a custom magnetic field profile for many purposes. The mostcommon is for strong holding applications, such as in robotics,manufacturing, or security. Other applications are for levitation, suchas in maglev trains. Yet other applications may utilize arrays fornoncontact magnetic gears. The technology disclosed herein is unique inthat target actuation in not used, but rather the detection of forces onsuch permanent magnets/arrays, at micro and nano scales.

Electrocardiogram products are ubiquitous. These technologies involvethe contact of electrodes with skin to detect voltage potentials. Thesepotentials are proportional to ionic currents generated by the heartduring contraction. Such information is used for screening, diagnosis,performance, and more. The most precise and comprehensive product is theHolter monitor, which typically has the most leads. Body surface mappingalso exists, but the large number of electrodes is generally impracticalfor any applications beyond research. Recent trends show miniaturizationof such technologies into two-lead patches over the heart. Wearablewatches are also entering this space, with limited diagnosticcapability. Products with FDA clearance are emerging related todetection of magnetic fields of the heart, improving triage of patientswith chest pain and accuracy of diagnosis. Yet, the technology isgreatly limited by the necessity for large instruments with magneticshielding and specialized facilities.

SUMMARY OF THE TECHNOLOGY

In brief summary, disclosed herein is methodology for detection ofsensitive magnetic fields with enhanced immunity from magneticinterference. The subject technology disclosed herein meets the needsdescribed above by enabling miniaturized magnetic sensing in portableand wearable equipment, while improving accuracy of existing sensors.

In at least one aspect, the subject technology relates to a gradiometerhaving a first magnet and a second magnet mechanically coupled togetherand aligned along a polarization axis. The first magnet and the secondmagnet are positioned such that a pair of like magnetic poles of thefirst magnet and the second magnet face in opposite directions, whereinthe first magnet and the second magnet are configured to move along thepolarization axis in response to a magnetic field. A sensing system isconfigured to measure a change in a gradient of the magnetic field basedon the movement of the first magnet and second magnet along thepolarization axis in response to the magnetic field.

In some embodiments, the change in a gradient of the magnetic field is asecond order gradient. In some cases, the change in a gradient of themagnetic field is a higher-than-second order gradient. In someembodiments, the gradiometer includes a third magnet mechanicallycoupled to the second magnet on a side opposite the first magnet to movealong the polarization axis, wherein the third magnet is positioned suchthat a pair of like magnetic poles of the third magnet and the secondmagnet face in opposite directions, wherein the change in a gradient ofthe magnetic field is a third order gradient.

In some embodiments, the first magnet and the second magnet arepositioned with a separation distance of substantially 2 mmtherebetween. In some cases, the sensing system is a microscope, and thechange in gradient is determined based on a change in deflection of thefirst magnet and/or the second magnet along the polarization axismeasured by the microscope. In some embodiments the first magnet and thesecond magnet are configured to move in mechanical resonance.

In at least one aspect, the subject technology relates to a method ofdetermining a change in a gradient of a magnetic field. The methodincludes positioning and mechanically coupling a first magnet and asecond magnet such that they move together along a polarization axis inresponse to a magnetic field. The first magnet and the second magnet arealigned along the polarization axis with like magnetic facing inopposite directions. Next, the method includes measuring, with a sensingsystem, a change in the gradient of the magnetic field based on themovement of the first magnet and the second magnet along thepolarization axis in response to the magnetic field.

In some embodiments, the change in the gradient of the magnetic field isa second order gradient and in some cases the change in the gradient ofthe magnetic field is a higher-than-second order gradient. The methodcan include mechanically coupling a third magnet to the second magnet ona side opposite the first magnet to move together with the second magnetalong the polarization axis. The third magnet can be positioned suchthat a pair of like magnetic poles of the third magnet and the secondmagnet face in opposite directions. The change in the gradient of themagnetic field can be a third order gradient.

In some embodiments the first magnet and the second magnet arepositioned with a separation distance of substantially 2 mmtherebetween. The sensing system can be a microscope and measuring thechange in the gradient of the magnetic field can include determining,with the microscope, a change in deflection of the first magnet and/orthe second magnet along the polarization axis, wherein the change in thegradient of the magnetic field is determined based on the change indeflection.

In at least one aspect, the subject technology relates to a method ofassembling a magnet array for a gradiometer. The method includesattaching a first magnet to a top surface of a first transparentsubstrate, the first transparent substrate having a greater width thanthe first magnet. A second magnet is attached to a top surface of asecond transparent substrate, the second transparent substrate having agreater width than the second magnet. The second magnet is positionedover the first magnet along a dipole axis. The first magnet is alignedwith the second magnet along the dipole axis by viewing the first magnetand second magnet with a microscope from a position directly above thesecond magnet along the dipole axis and adjusting the position of thefirst magnet and/or the second magnet. Epoxy is placed on top of thefirst magnet. The second magnet is lowered onto the first magnet whilecontinuing to view the first magnet and the second magnet with themicroscope to maintain alignment along the dipole axis.

In some embodiments, the epoxy is UV curable epoxy which is cured withUV radiation to attach the first magnet and the second magnet. The firstmagnet is then held to the first transparent substrate with cured UVepoxy and the second magnet is held to the second transparent substratewith cured UV epoxy. In some cases, the first transparent substrate is aglass coverslip and the second transparent substrate is a glasscoverslip.

In some embodiments, after the step of lowering the second magnet ontothe first magnet, the method includes: curing the epoxy to attach thefirst magnet to the second magnet; breaking and removing the firsttransparent substrate from the first magnet; and breaking and removingthe second transparent substrate from the second magnet. In some cases,the second magnet is attached to the second transparent substrate with alike polar orientation to the first magnet. The first magnet can besubstantially (e.g. +/−10%) the same size as the second magnet.

BRIEF DESCRIPTION OF THE DRAWINGS

So that those having ordinary skill in the art to which the disclosedsystem pertains will more readily understand how to make and use thesame, reference may be had to the following drawings.

FIGS. 1 a, 1 b, 1 c, and 1 d show magnet arrangements for gradiometersin accordance with the subject technology.

FIG. 2a is a diagram of forces on an assembly of magnets configured inaccordance with the subject technology.

FIG. 2b is a second order gradiometer configured in accordance with thesubject technology.

FIG. 3a is a guinea pig heart in Langendorf preparation as part of anexperimental setup for measuring forces and gradients.

FIGS. 3b-3d are graphs of the data obtained from the experimental setupof FIG. 3 a.

FIG. 4 is a series of magnets with varying thickness for use as part ofthe systems disclosed herein.

FIGS. 5, 6, and 7 are graphs of the data obtained from a gradiometerutilizing the magnet geometries of FIG. 4.

FIG. 8 is a series of magnet pairs with varying thickness for use aspart of the systems disclosed herein.

FIGS. 9-10 are graphs of the data obtained from a gradiometer utilizingthe magnet pair geometries of FIG. 8.

FIGS. 11a , 11 b, 11 c, 11 d, 11 e, and 11 f are exemplary steps of amanufacturing process for assembling a magnet assembly in accordancewith the subject technology.

FIGS. 12a, 12b are overhead views of a magnet assembly in accordancewith the subject technology.

FIGS. 12c, 12d are overhead views of another magnet assembly inaccordance with the subject technology.

FIG. 13 is a schematic diagram of various configurations of magnetassemblies in accordance with the subject technology.

FIGS. 14a, 14b, and 14c are graphs of data obtained using one of themagnet assemblies of FIG. 13.

FIGS. 15a, 15b, and 15c are graphs of data obtained using another one ofthe magnet assemblies of FIG. 13.

FIGS. 16a, 16b, and 16c are graphs of data obtained using yet anotherone of the magnet assemblies of FIG. 13.

FIG. 17 is a schematic diagram of a magnet assembly in accordance withthe subject technology.

FIGS. 18a, 18b, and 18c are graphs of data obtained using the magnetassemblies of FIG. 17.

FIG. 19 is a schematic diagram of a gradiometer measuring a biomagneticsource, in accordance with the subject technology.

FIG. 20 is a schematic diagram of another gradiometer measuring abiomagnetic source, in accordance with the subject technology.

DETAILED DESCRIPTION

The advantages, and other features of the systems and methods disclosedherein, will become more readily apparent to those having ordinary skillin the art from the following detailed description of certain preferredembodiments taken in conjunction with the drawings which set forthrepresentative embodiments of the present invention. Like referencenumerals are used herein to denote like parts. Further, words denotingorientation such as “upper”, “lower”, “distal”, and “proximate” aremerely used to help describe the location of components with respect toone another. For example, an “upper” surface of a part is merely meantto describe a surface that is separate from the “lower” surface of thatsame part. No words denoting orientation are used to describe anabsolute orientation (i.e. where an “upper” part must always at a higherelevation).

The subject technology overcomes many of the prior art problemsassociated with sensing systems, and particularly overcomes problemsfound in typical force sensors. In brief summary, disclosed herein ismethodology for detection of sensitive magnetic fields with enhancedimmunity from magnetic interference. All of today's magnetic sensorsdetect magnetic intensity, which means they are sensitive to all formsof interference, such as the Earth's magnetic field and expansiveelectromagnetic infrastructure. A large span of techniques have beendeveloped to increase immunity to interference, such as differentialsensing with multiple sensors, cryogenic cooling, large magneticallyshielded rooms, common mode rejection electronics, and more. The systemsand methods disclosed herein enable the measurement of sensitivemagnetic fields at a single point, with a single sensor, which haveincreased immunity to magnetic interference. This is achieved by thedirect detection of spatial derivatives of the magnetic field, where asensor becomes more sensitive to objects in close proximity, and farless sensitive to sources that are farther away.

This technology disclosure builds upon single-point gradiometers (e.g.as disclosed in U.S. Provisional Patent App. No. 63/159,692,incorporated herein by reference) that selectively measure the firstorder spatial gradient of a magnetic field. Included herein is adiscussed of the rationale for the measurement of higher order gradientswith single-point gradiometry. Exemplary systems focused on second ordergradiometry are disclosed herein in detail, supported by fabricationmethods, a measurement system, and results demonstrating the selectivemeasurement of a second order gradient (with immunity to magneticintensities and first order gradients). The benefits described for firstorder gradiometers are enhanced with these designs: greater immunity tonoise and interference, less post-processing, a path toward a morecost-effective product. The greatest benefit this class of gradiometersoffers to sensitive magnetometry is unshielded sensing in ambientenvironments, a direct product of immunity to magnetic interference. Thetechnology disclosed herein builds on prior technology that detectsmagnetic field intensities, and instead measures various gradients,allowing for improved accuracy and miniaturization.

Magnetic sensing applications are ubiquitous, especially for highresolution sensors. These applications span medical to consumer, toindustrial. In the medical space, leading applications may be in cardiacand cancer imaging. For those applications, the subject technologyprovides an array of sensors which produce an image of magnetic fieldswithout contact with the patient. In the cardiac case, the magneticfields describe ionic currents of the heart, analogous to EKGs, but with3D information and gathered without contact. For cancer imaging, themagnetic fields describe the activity of injected nanoparticles, whichidentify solid cancerous tumor margins for the purpose of staging andsurgically removing cancerous tissue.

In consumer products, iPhones, automobiles, laptops, and the like allcontain magnetic sensors. The technology disclosed herein is compatiblewith semiconductor processes, allowing for integration in those devices.The disclosed sensors may replace existing sensors in some cases, and inother cases may supplement them. The latter is possible as the sensorsdisclosed herein are directly sensing gradient and higher order gradientmagnetic fields, rather than field intensities. This provides valuableinformation with reduced interference that may aid in enhancedmeasurement precision in such devices. Measurements may be targeting theEarth's field, rotary components, proximity to magnetic/ferromagneticcomponents, battery health, or other systems which can be measuredthrough gradients of magnetic fields.

Overall, the technology disclosed herein allows for the detection ofspatial gradients of magnetic fields. One may use existing literature ormagnetic sensors to measure the spatial profile of a magnetic field froma source, such as that of a human heart or of a ferromagnetic pipeline.Most magnetic field profiles contain many spatial gradients alongseveral axes. The simplest imagination is along the central, easy axisof the source, where one may measure the first order spatial gradient(dB/dx) or the second order spatial gradient (d2B/dx2). Furthermore,combinations of different magnetic arrays and/or sensors would enablesensitivity to a variety of spatial gradients, such as d(Bz)/dx, whichmay be an intuitive measurement for interpreting images of the magneticfields of the heart.

Referring now to FIGS. 1a -1 d, various magnet arrangements forgradiometers are shown. The majority of magnetic sources (includingbiomagnetic sources) produce magnetic fields that decay with increasingdistance from the source as function of 1/d³. These are spatial magneticgradients. First order gradiometers directly measure the gradient of themagnetic field and have reduced sensitivity to noise from magneticintensity. Disclosed herein are second (and higher) order gradiometerswhich have reduced sensitivity to both first order gradients andintensities. The immunity to such fields will significantly decreasesensitivity to magnetic interference and noise. Due to the small size ofthe sensing platforms disclosed, the disclosed technology is able to getmuch closer to a sample, where second order gradients are at theirmaximum. These devices will similarly feel a force on a single axisdirectly proportional to the magnitude of the second order, or higherorder, gradient magnetic field.

FIG. 1a shows a simple magnet 102, which would feel a force proportionalto the first order gradient of a magnetic field. FIG. 1B shows astructure 105 with two magnets 104, 106 in a “back-to-back”configuration, and with like poles adjacent to (and facing in oppositedirections) one another. The structure 105 would feel a forceproportional to a second order gradient of a magnetic field. FIG. 1cshows a structure 109 with a combination of three magnets 108, 110, 112,where the central magnet 110 has like poles adjacent the outer magnets108, 112 (with adjacent magnetic poles facing in opposite directions).Using the system 109 of FIG. 1 c, the 3rd order gradient of a magneticfield can be measured directly. Force measurements can be taken inaccordance with the following:

$\begin{matrix}{F_{1M} = {M \cdot \frac{dB}{dx}}} & (1)\end{matrix}$ $\begin{matrix}{{F_{2M} = {( {{M \cdot \frac{dB}{{dx}_{1}}} - {M \cdot \frac{dB}{{dx}_{2}}}} ) \propto M}}{\cdot \frac{d^{2}B}{{dx}^{2}}}} & (2)\end{matrix}$ $\begin{matrix}{{F_{3M} = {( {{M \cdot \frac{d^{2}B}{{dx}_{1}^{2}}} - {M \cdot \frac{d^{2}B}{{dx}_{2}^{2}}}} ) \propto M}}{\cdot \frac{d^{3}B}{{dx}^{3}}}} & (3)\end{matrix}$

In Eqns. 1-3, F is a force on the structure, M is the moment of a singlemagnet (all magnets 102, 104, 106, 108, 110, 112 are identical), B isthe magnetic intensity, and x is the position axis common to the magnetdipoles.

FIG. 1d shows a system 120 which produces great sensitivity to magneticfield intensities. A single magnet 116 will also experience a torquefrom magnetic field intensities. Here, a magnet 116 is embedded in aplate 114 which will produce rotation of the plate 114 around a centralaxis 118 upon feeling torque. A sphere 122 can be employed to measurethis torque by measuring the voltage between the plate 114 and sphere122. At far separations (>1 micron), the voltage will largely beproportional to electrostatic forces, scaling as 1/d².

Referring now to FIG. 2 a, a diagram 200 shows forces on back-to-backmagnets 202, 204 (i.e. with like poles positioned adjacent to oneanother). FIG. 2a is illustrative of another benefit of a second ordergradiometer design. By rigidly coupling two back-to-back magnets 202,204 together, we can expect cancelation of torsional forces 206. For thefirst order gradiometer we use anisotropic design of microscale forcesto minimize the effect of torsion. For a higher order gradiometerdesign, further attenuation of torsional forces will be enhanced by thestructure. The equations 4-5 below demonstrate this, where T is atorsional force, M is the magnetic moment (all magnets are identical), Bis the magnetic field intensity, and θ is the degrees of misalignmentbetween the magnetic moment vectors and the magnetic field vector.

T _(1,M) =M×B=MBsin θ  (4)

T _(2,M)=(MB sin θ+MB sin (θ−180))=MB(sin θ+sin (θ−180))=0  (5)

In equation 4, one may note that if θ=0 (the magnet moment vector isaligned with the magnetic field vector), then the torque is zero.However, in the equation for two back to back magnets, this is alwayszero, independent of θ.

Referring now to FIG. 2 b, a system 210 is shown which illustrates apractical device for the sensitive measurement of second order gradientmagnetic fields as disclosed herein. The system 210 includes twoback-to-back magnets 212, 214 mechanically coupled together (e.g.attached or rigidly connected) at like poles. A microassembly 216 allowsthe attached magnets 212, 214 to move in either direction along apolarization axis 222 of the magnets 212, 214. A proof mass is includedfor reference, and the system 210 is connected to a sensing system 220,which can be a MEMS force transducer or other system designed to measureforce. It should be understood that while two magnets 212, 214 are shownin the system 210 of FIG. 2 b, other the other arrangements of magnetsshown herein could also be employed as part of the system 210 instead ofthe magnets 212, 214.

Generally, sensors make gradiometric measurements via the subtraction of2 sensors, separated by some distance along the axis of measurement. Formany applications, this is a disadvantage because the sensors are spacedso far apart that the further away sensor barely sees any field, and maysee only background. Furthermore, the subtraction of two signals can addnoise and complexity to a measurement. However, there are disadvantagesto being too small as well. The signal has higher order gradients tooand so a gradient measurement is not a perfect representation of thesignal. In accordance with the teachings herein, optimum magnet designcan help amplify the sensitivity. A recent paper in Nature scientificreports used a research laboratory setup with an atomic magnetometer tomeasure the profile of a cardiac magnetic field extending away from theheart. This data is relied on herein in further calculations and designoptimization routines for more credible accuracy.

Referring now to FIG. 3 a, the heart of a guinea pig 302 is shown in aLangendorf preparation, an experimental setup that simulates a normalphysiological environment. FIGS. 3b-3d are graphs 304, 306, 308 showingmeasurements presented in the journal article, with data points shown ingraph 304 as dots 310. The peak-to-peak of a magnetic intensity signalfrom the heart is measured. Borrowing language from electrocardiogrammeasurements, this would be a measurement of the QRS complex. Therefore,the y axis is labeled QRSpp, and the units are picoTesla. At 6 mm awayfrom the heart, about 100 pT is measured. As the sensor is moved furtheraway, the signal diminishes. The fit to this profile is described asfollowing the form of equations 6-7 below, with other publicationshaving found similar profiles.

$\begin{matrix}{B_{cardiac} \propto \frac{1}{x^{1.7}}} & (6)\end{matrix}$ $\begin{matrix}{B_{magnet} \propto \frac{1}{x^{3}}} & (7)\end{matrix}$

This relationship is noted herein as the profile of the magnetic fieldextending from a cube permanent magnet will closely follow a cubicprofile. Therefore, the cardiac signal will have smaller gradients thana permanent magnet, but the tradeoffs still provide opportunity. Thisprior work provides a foundation for understanding the systems describedin the following discussions. Finally, the reason for the difference infield profile is directly due to the geometry of the heart, includingcontributions from ionic currents throughout.

Referring again to FIG. 3 b, the graph line 312 is fit to the datapoints 310, with an exponent of 1.7. With this field profile we cancompute the second order spatial gradient along this profile, yieldingthe curve 314 shown in FIG. 3 c. Here, we note that a 200 pT/cm field isanticipated at about 7 mm away from the heart, and the signal falls offa bit more quickly. Furthermore, we compute the second order gradientgiven by graph line 316 in FIG. 3d by taking the spatial derivativeagain. Here, we anticipate a field of 700 pT/cm² at about 8 mm from theheart. And again, the signal 316 falls off more quickly as you move awayfrom the heart.

Of note as well is the estimated magnitude of magnetic noise fromelectromagnetic equipment in hospitals (refrigerators, MRI machines,etc.). In most areas of a hospital, there is expected to be about 50 pTof noise, threatening to obscure a biomagnetic measurement. This isprecisely the reason why today's biomagnetic technology requiresmagnetic shielding. Such shielding is sometimes implemented as aspecialized facility, where thick steel or Nickel alloys are integratedinto the walls of a room to attenuate magnetic interference. In othercases, a large cylinder with a bore to fit an entire human being andsensing system is purchased for a room.

These are the state of the art for biomagnetic imaging. In each graph304, 306, 308, noise level is denoted by graph line 320. In graph 306 ofFIG. 3 c, we note that the noise levels 320 are smaller, relative to thesignal 314. This is even further enhanced in graph 308 of FIG. 3 d. Thesignal to noise ratios at closest proximity are 2, 4, and 14respectively, across the three graphs 304, 306, and 308. One can nowclearly see the advantage of measuring gradient fields when close to thesource of interest and farther away from sources of interference. Usingthe published cardiac data from a biomagnetic sensor, we proceed withour optimization discussion.

Referring now to FIG. 4, images magnets 400 a, 400 b, 400 c, 400 d, 400e (generally 400) with varying thickness along their dipole axis, whileconserving volume. Therefore, the magnetic moment is also constant. Thischanges the interaction of the magnet with the field because a largerspatial region of the field profile is interacting with the magnet. InFIG. 4, we illustrate the design of such magnets, where the nominaldesign (magnet 400 b) is a cube of thickness 0.25 mm. The otherthicknesses are as follows: magnet 400 a is a thickness of 0.125 mm;magnet 400 c is a thickness of 0.5 mm; magnet 400 d is a thickness of 1mm; and magnet 400 e is a thickness of 2 mm. All dimensions shown aremanufacturable. The magnets 400 can be used within the gradiometersdescribed herein.

Referring now to FIG. 5, a graph 500 represents an extrapolated fit fromthe cardiac data up to 2 mm from the source. This is reasonable as themagnetic sensor chip used in the systems disclosed herein, including theASIC, MEMS, and magnet array, is on a 2 mm by 2 mm die, and thus has adistance of 1 mm from the edge to the center of the magnet array. At 2mm, we now observe estimated field magnitudes near 5000 pT/cm, or 5nT/cm. Now, we evaluate the gradient magnetic field that would beobserved by each of the magnet geometries from FIG. 4. On this plot 500,it appears that the fields are somewhat similar. However, there ismarginal gain that is calculated and plotted in graph 600 of FIG. 6. InFIG. 6, the x-axis is the thickness of the magnet (here labeled asseparation). The y-axis is the percentage gain in force (or gradientmagnetic field) experienced by the magnet. The different sets of datapoints 602, 604, 606 are represented by dashed lines 502, 504, 506,respectively, in FIG. 5 (which are at 3 mm, 4 mm, and 7 mm from thesource, respectively). Notably, a magnet with a 2 mm length willexperience up to 30% gain in force when positioned 3 mm from the source.This translates, in the systems disclosed herein, to a device using thesame materials in different geometries, which has enhanced sensitivityto gradient magnetic fields.

Referring now to FIG. 7, a graph 700 illustrates the reason for thiseffect. The dashed line 702 represents the center of the device, placedsome distance from the source. Graph line 704 represents the gradientmagnetic field observed by a 2 mm long magnet and graph line 706represents a gradient field experienced by a 0.5 mm magnet. Here, we canobserve by eye that the 2 mm magnet would experience a greater signal.It is noteworthy that there is a sweet spot for such a measurement. Onecould make the magnet arbitrarily long and continue to increase thegain, but the greatest gain occurs close to the source. In order to getclose to the source, one needs a smaller magnet and a smaller device.

Next, we can extrapolate our methods from optimizing the first ordergradient to an optimization of a second order gradient sensor. Referringnow to FIG. 8, a similar arrangement to FIG. 4 is shown, with magnets ofvarying thickness. The difference is that each array 800 a, 800 b, 800c, 800 d, 800 e of FIG. 8 is a combination of two magnets positionedback-to-back (e.g. with like poles adjacent). Recall that the force on astructure here is proportional to the second order spatial gradient ofthe magnetic field along the direction of the common dipole axis of themagnets.

Referring now to FIG. 9 a graph 900 similar to graph 500 of FIG. 5 isshown. To create graph 900, we computed second order gradient field upto 2 mm from the source, calculated from the published cardiac data.Each of the profiles represents a different magnet array thicknessillustrated in FIG. 8 as follows: graph line 902 corresponds with athickness of 2 mm; graph line 904 corresponds with a thickness of 1 mm;graph line 906 corresponds with a thickness of 0.5 mm; and graph line908 corresponds with a thickness of 0.25 mm. This time, the termseparation may make more sense since this array will be made from twoback-to-back magnets. Notably, the profiles are much more visiblyseparated in graph 900, as compared to graph 500. This indicates thatthis configuration is more sensitive to the thickness variable, wheregreater thicknesses yield larger field measurements. This is anextrapolation of the concept illustrated in graph 700 of FIG. 7, as thespatial derivative of the plot was calculated a second time. Threeslices (dotted lines 910, 912, 914) from graph 900 are compared in graph1000 of FIG. 10, with slice 910 corresponding to graph points 1002,slice 912 corresponding to graph points 1004, and slice 914corresponding to graph points 1006. We observe that thickness of theassembly (separation between the magnets) has an exponentialrelationship with the sensitivity, illustrated here as a force gainpercentage. Notably, an array with a 2 mm separation between the magnetswould have a 7000% gain in the force. This provides an effective guidefor a MEMS second order gradiometer.

Even with this design understood, fabrication of the array presentsserious design challenges. Therefore we now discuss custom proceduresfor fabrication of a magnetic array and corresponding system inaccordance with the subject technology.

Firstly, we should note the obvious: magnets in a back-to-back positionwill impose a strong force on each other, and that force will increaseexponentially as you bring them closer together. When the dipoles areperfectly aligned, the force should only be on that axis and should beeasy to counter. However, small degrees of misalignment will createlarge off-axis forces. Secondly, the magnets are small (0.25 to 0.5 mm)and so manipulation of them is challenging.

We first considered fabricating a mold to hold each of the magnets,using, for example, miniaturized equipment, including a small drillpress, end mill, PCB cutting saw, and other common fabrication tools.However, the magnets are cubes and it is challenging, or impossible, tomake a perfect 90 degree cut in a mold, since typically circular motionof a drill is used. The smallest drill sizes that we are commerciallyavailable are around 0.1 mm. This limits our ability to fabricate a holein a part that could properly constrain the magnet.

Next, we considered 3D printing, where one can obtain lateralresolutions of about 25 microns using direct laser writing in acommercial foundry. However, these materials are not transparent, and sothe ability to align the magnet dipoles under a microscope would bechallenging. We considered precision lithography approaches, such asspinning photoresist and using a mask aligner to precisely bring the twomagnets together, but such a process may be long and cumbersome. Thismight be considered for higher throughput fabrication of the arrayshowever.

After considering the constraints, we developed a fabrication processusing microfabrication equipment consisting of articulated vacuummicropipettes, a probe station with a microscope, and micromanipulatorswith 7 degrees of freedom. This process is shown in FIGS. 11a-11f anddescribed in detail below.

To that end, referring now to FIGS. 11a 11 f, exemplary steps of amanufacturing process for a magnet array in accordance with the subjecttechnology are shown. The process begins in FIG. 11 a, where a 70 umthin clear glass coverslip 1102 is suspended. Notably, the coverslip1102 could be replaced by another substrate in other cases. A large,external magnet 1104 is placed below the coverslip 1102. A microscaledrop of UV curable epoxy 1106 is dotted on the coverslip 1102 using aprobe. A micromagnet 1108 is introduced from above. The micromagnet 1108is manipulated by a vacuum pick-and-place micropipette. This step couldbe completed with tweezers (or other instruments) as well, since theexternal magnet 1104 below will guide and center the micromagnet 1108.The external magnet's 1104 dipole orientation is known and so we knowthe external magnet 1108 will align with it on the coverslip 1102. TheUV curable epoxy 1106 is cured with 30 s of UV exposure and thecoverslip 1102 with micromagnet 1108 is removed by moving up along thedipole axis 1110 to minimize transverse forces.

This step is then repeated with a second magnet 1112 (which can includea similar magnet and coverslip of similar sizes to the first magnet 1108and coverslip 1102), as shown in FIG. 11 b. One of the assemblies 1114 bis then rotated 180 degrees. The bottom assembly 1114 a is fixed to aflat, stationary surface. A microscale drop of UV glue (1124 of FIG. 11d) is added to the bottom magnet 1108. The top assembly 1114 b is fixedto a flat bottom piece on a micromanipulator.

As shown in FIG. 11 c, a brightfield microscope 1116 is used to focus onthe bottom micromagnet 1108. The top micromagnet 1112 is kept about 1 cmhigher than the bottom magnet 1108 as it is translated to align thedipole axis 1110. Since the glass coverslips 1102, 1118 are transparent,we can observe both micromagnets 1108, 1112 through the microscope 1116.The microscope 1116 objective is translated up and down (e.g. along theaxis 1110) to improve alignment of the magnet dipoles. The topmicromagnet 1112 is slowly lowered toward the bottom micromagnet 1108.The magnets 1108, 1112 are reflective, and so any small change in thereflected light indicates contact between the magnets 1108, 1112 as theyare lowered, yielding the assembly 1122 of FIG. 11 d. It is importantthe glass coverslips 1102, 1118 are much wider than the magnets 1108,1112, as this makes the counteraction of torsional forces between themagnets 1108, 1112 easier. The magnets 1108, 1112, are held back-to-backas the UV epoxy 1124 is cured with 30 s of UV radiation. The assemblymay be baked overnight at 60 C to ensure a full cure of the epoxy.

Referring now to FIG. 11 e, a diamond tipped pen (not shown distinctly)is then introduced directly on top of the magnet 1112 face, and pressed.The glass coverslip 1118 will cleanly crack without disturbing themagnets 1108, 1112. The assembly 1126 is then flipped 180 degrees andthis is repeated on the bottom glass coverslip 1102. The final assembly1128 is shown in FIG. 11 f. This assembly 1128 (i.e. magnet array) canthen be manipulated via the vacuum pick-and-place micropipette andassembled on other structures for characterization. For example, theassembly 1128 can be used as part of a gradiometer as disclosed herein.This can build on existing techniques for fabrication on post-releaseMEMS accelerometers used to detect ultralow forces on the magnets.

We utilize an elastomer and microscope system to characterize forces onthe micromagnet arrays (e.g. assembly 1128). A polydimethylsiloxanemicropost is fabricate by pouring the mixture into a 3D printed mold andbaking overnight. The post dimensions are approximately 500×500 um incross section and 800 microns tall. Microdroplets of UV epoxy are addedto the top of the posts and the magnet assembly 1128 is fixed at the topof the post. An external electromagnetic coil system is introduced oneither side of the assembly 1128 to impose magnetic fields. Themicroscope will measure deflections of the post via image processing,and forces will be inferred. Therefore the gradiometer uses acombination of the magnet array of the assembly 1128, with the change ingradient being measured by the microscope acting as the sensing system.

Referring now to FIGS. 12a -12 d, two sets of magnet assemblies 1202,1204 are shown. In FIGS. 12a -12 b, the magnet assembly 1202 includestwo micromagnets 1206, 1208 with aligned dipoles, simply holdingthemselves together by magnetic forces. In FIGS. 12c -12 d, two magnets1210, 1212 are positioned in a back-to-back configuration, fabricated bythe procedure shown in FIGS. 11a -11 f. FIGS. 12a and 12c include partof the coil 1214 for scale. Only the iron core of the coil 1214 isvisible in the images, where the coil diameter is approximately 3 timeslarger (approximately 2.5 cm diameter and 4 cm long). A DC current willbe controlled in the coil(s) 1214 to impose a magnetic force. Forsimplicity, we will refer to the aligned-pole magnets as “NS” magnetsand to the back-to-back magnets as “NN” herein.

Referring now to FIG. 13, various configurations of magnet assemblies1302, 1304, 1306. Each assembly 1302, 1304, 1306 depicts one arrangementwith a magnet pair 1320 in an “NS” arrangement and a second arrangementwith a magnet pair 1320 in an “NN” arrangement (notably, the NS and NNarrangements would be deployed independently, not as part of the samegradiometers). The first assembly 1302, SC_0d, is a single coil 1308aligned at 0 degrees with the magnet dipole axis. The spatial magneticfield profile from a single coil contains gradients of 0^(th)(intensity), first, and second orders. The alignment of the magnets 1320is expected to either pull or push the assembly 1302. The secondassembly 1304, SC_90d, is a single coil 1310 at 90 degrees relative tothe dipole axis. The extreme misalignment is expected to impose a torqueon the magnets 1320. The third assembly 1306, DC_0d, is a dual coil 1312system aligned at 0 degrees with the dipole axis. The coils 1312 may bewired such that they produce magnetic field in the same direction(parallel coils) or in opposite directions (anti-parallel coils). In theparallel case, the spatial magnetic field contains only magnetic fieldintensities and no gradients. In the anti-parallel case, the spatialmagnetic field profile contains first and second order gradients and nointensities. We will only characterize the latter (anti-parallel) casesince the first configuration, given by assembly 1302, alreadyencompasses on-axis magnetic field intensities.

The experimental procedure is to sweep the current in the coils from anegative current to a positive current, recording microscope images ateach current value. The current is directly proportional each of the0^(th), first, and second order fields in the coils (the fieldprofile/shape is preserved and is only scaled by the current). Theimages are post-processed to record deflection in X (coil axis), Y (90degrees from coil axis in plane), and 0 (degree of rotation in XYplane). Out-of-plane motion is not accounted for and is considered to beminimal for small deflections of the PDMS post. It is noteworthy thatthe PDMS post is very soft in all directions compared to forces weimpose on the magnets, making this an appropriate characterizationplatform.

Referring now to FIGS. 14a -14 c, graphs 1400 a, 1400 b, 1400 c of theresults for the first assembly 1302 are shown. Graph points 1402represent the assembly 1302 when the magnets are positioned in an “NS”arrangement, while graph points 1404 represent the assembly 1302 whenthe magnets are positioned in an “NN” arrangement. Graph 1400 arepresents deflection along the x axis, graph 1400 b representsdeflection along the y axis, and graph 1400 c is θ deflection. Comparedto the NS assembly (e.g. points 1402), the NN assembly (points 1404)reduced the X deflection by 34x. We note that the deflections becomenonlinear at larger currents, as the field profile from a single coilcontains 0^(th), first, and second order gradients (it is nonlinear).The NN assembly reduces Y deflections by 13x, and we observe a largernonlinear effect in the NS deflection profile, partly from the nonlinearfield and partly from misalignment of the magnet fabrication. Finally,the NN magnets reduce 0 deflection by about 2x compared to the NSassembly. Overall, we interpret this as decreased sensitivity to 0^(th)and first order gradient fields, which will improve even further withgreater fabrication alignment.

It is noteworthy that is the only configuration that presents secondorder gradient fields on-axis with the micromagnet assemblies. Thesecond order gradient component is small compared to the 0^(th) andfirst order gradients, especially for these electromagnets. The Xdeflection, supplemented by the results of FIGS. 15 and 16 describedbelow, demonstrate that the NN configuration is selectively sensitive toforces from second order gradient magnetic fields.

Referring now to FIGS. 15a -15 c, graphs 1500 a, 1500 b, 1500 c of theresults for the second assembly 1304 are shown. Graph points 1502represent the assembly 1304 when the magnets are positioned in an “NS”arrangement, while graph points 1504 represent the assembly 1304 whenthe magnets are positioned in an “NN” arrangement. Graph 1500 arepresents deflection along the x axis, graph 1500 b representsdeflection along the y axis, and graph 1500 c is θ deflection. Comparedto the NS assembly, the NN assembly reduced deflections in the Xdirection by 8x, in the Y direction by 84x, and in the θ direction by45x. This serves as validation of the torsional equations 4-5 discussedearlier. This is a reminder that a permanent magnet, used as a sensor,is sensitive to both magnetic intensities (as a torque) and 1^(st) ordergradients (as a force along the dipole axis). Our soft pillar platformillustrates this quite well. Our fabrication of a first ordergradiometer took advantage of anisotropic mechanical structures on MEMSaccelerometers to decrease sensitivity to torsional forces, but it isstill a remaining source of noise. The graphs 1500 a, 1500 b, 1500 cshow that the NN assembly selectively reduces sensitivity to both 0^(th)and first order gradients by design.

Referring now to FIGS. 16a -16 c, graphs 1600 a, 1600 b, 1600 c of theresults for the third assembly 1306 with the antiparallel coils areshown. Graph points 1602 represent the assembly 1306 when the magnetsare positioned in an “NS” arrangement, while graph points 1604 representthe assembly 1306 when the magnets are positioned in an “NN”arrangement. Graph 1600 a represents deflection along the x axis, graph1600 b represents deflection along the y axis, and graph 1600 c is θdeflection. Compared to the NS assembly, the NN assembly reduces Xdeflection by 175x, the Y deflection by 2x, and the θ deflection by 10x.This demonstrates that the NN assembly is highly effective at cancelingeffects of first order gradient forces. It is noteworthy as well thatthe Y and θ deflection are very small for both assemblies. This issomewhat anticipated, because an antiparallel configuration imposes onlyfirst order gradient fields without 0^(th) or second order gradientfields.

Referring now to FIG. 17, further experimentation was done on a fourthexemplary magnet assembly 1700 which uses the configuration of assembly1302. On the left, the assembly 1700 is shown with a magnet pair 1702 inan “NS” arrangement while on the right the assembly is shown with themagnet pair 1702 in an “NN” arrangement. In this instance, we sweep theposition of the magnet pair 1702 and keep a constant current from a coil1704.

Deflection of the magnet pairs 1702 is recorded for analysis in thegraphs 1800 a, 1800 b, 1800 c of FIGS. 18a -18 c. Graph points 1802represent the assembly 1700 when the magnets are positioned in an “NS”arrangement, while graph points 1804 represent the assembly 1700 whenthe magnets are positioned in an “NN” arrangement. Graph 1800 arepresents deflection along the x axis, graph 1800 b representsdeflection along the y axis, and graph 1800 c is θ deflection. Thespatial profiles of Y and θ deflections are similar and small. The NNarrangement appears to reduce deflection by 2x in both. Any deflectionsof the assemblies here is attributed to fabrication misalignment on theposts.

The spatial profile of X deflection for both assemblies is the mostinteresting. The NS assembly experiences a deflection that has anexponential relationship with the distance of the magnets 1702 from thecoil 1704. This is expected, as the coil 1704 contains higher ordergradients in its magnetic field profile. The X-deflection of the NNassembly is very different however. This NN assembly records a linearrelationship with distance from the coil. This indicates that thisassembly is not sensitive to 0^(th) and first order gradients, and italso indicates that there are extremely smaller third order and highergradients in this magnetic field profile, if any (3^(rd) order gradientswould make this relationship also look exponential). Therefore using ourmethods and electromagnetic system, the fabricated NN assembly isselectively sensitive to second order gradient magnetic fields.

It is noteworthy that some magnetocardiography articles have discussedthe utility of a three-dimensional gradient, d(dB/dz)dx, where z isorthogonal to the biomagnetic source and x is in a parallel plane. Thisvariable is directly proportional to the intuitive ionic current flow inthe heart and so directly sensing this variable may greatly minimizerequirements for post processing. In order to measure this quantitytoday, one would need information from 4 to 6 sensors. We propose asystem using two sensors FIG. 19, and a system using a single sensor inFIG. 20.

Using our unique single-point gradiometer designs, this variable of abiomagnetic source 1906 could be measured by the system 1900 illustratedin FIG. 19. The system 1900 includes two order gradiometers 1902, 1904next to each other on the same chip 1908 (which can include a sensingsystem, and other components of the gradiometer discussed herein). Thesevector sensors 1902, 1904 are oriented along an axis orthogonal to theheart. A subtraction between these sensors 1902, 1904 would yield thisvariable.

Referring now to FIG. 20, an alternative system 2000 is shown, whichwill feel a force proportional to a planar gradient Bz. In the system2000, we combine our abilities to fabricate unique micromagnet arrayswith our abilities to functionalize MEMS platforms with foreignmaterials structures. Similar to how back-to-back magnets that arerigidly fixed will cancel torsional and direct forces, it isstraightforward to carry these principles into the design of an array2002, for which a direct force (e.g. from a biomagnetic source 2004) isproportional to the variable of interest.

Overall, the disclosure herein provides a system and method formeasuring magnetic gradients of the second order or greater. In somecases, the force felt by the sensors disclosed herein will be directlyproportional to the magnetic field gradient.

All orientations and arrangements of the components shown herein areused by way of example only. Further, it will be appreciated by those ofordinary skill in the pertinent art that the functions of severalelements may, in alternative embodiments, be carried out by fewerelements or a single element. Similarly, in some embodiments, anyfunctional element may perform fewer, or different, operations thanthose described with respect to the illustrated embodiment. Also,functional elements) shown as distinct for purposes of illustration maybe incorporated within other functional elements in a particularimplementation.

While the subject technology has been described with respect topreferred embodiments, those skilled in the art will readily appreciatethat various changes and/or modifications can be made to the subjecttechnology without departing from the spirit or scope of the subjecttechnology. For example, each claim may depend from any or all claims ina multiple dependent manner even though such has not been originallyclaimed.

What is claimed is:
 1. A gradiometer comprising: a first magnet and asecond magnet mechanically coupled together and aligned along apolarization axis, the first magnet and the second magnet beingpositioned such that a pair of like magnetic poles of the first magnetand the second magnet face in opposite directions, wherein the firstmagnet and the second magnet are configured to move along thepolarization axis in response to a magnetic field; a sensing systemconfigured to measure a change in a gradient of the magnetic field basedon the movement of the first magnet and second magnet along thepolarization axis in response to the magnetic field.
 2. The gradiometerof claim 1, wherein the change in a gradient of the magnetic field is asecond order gradient.
 3. The gradiometer of claim 1, wherein the changein a gradient of the magnetic field is a higher-than-second ordergradient.
 4. The gradiometer of claim 1, further comprising a thirdmagnet mechanically coupled to the second magnet on a side opposite thefirst magnet to move along the polarization axis, wherein the thirdmagnet is positioned such that a pair of like magnetic poles of thethird magnet and the second magnet face in opposite directions, whereinthe change in a gradient of the magnetic field is a third ordergradient.
 5. The gradiometer of claim 1, wherein the first magnet andthe second magnet are positioned with a separation distance ofsubstantially 2 mm therebetween.
 6. The gradiometer of claim 1, whereinthe sensing system is a microscope, and the change in gradient isdetermined based on a change in deflection of the first magnet and/orthe second magnet along the polarization axis measured by themicroscope.
 7. The gradiometer of claim 1, wherein the first magnet andthe second magnet are configured to move in mechanical resonance.
 8. Amethod of determining a change in a gradient of a magnetic field,comprising: positioning and mechanically coupling a first magnet and asecond magnet such that they move together along a polarization axis inresponse to a magnetic field, the first magnet and the second magnetaligned along the polarization axis with like magnetic facing inopposite directions, measuring, with a sensing system, a change in thegradient of the magnetic field based on the movement of the first magnetand the second magnet along the polarization axis in response to themagnetic field.
 9. The method of claim 8, wherein the change in thegradient of the magnetic field is a second order gradient.
 10. Themethod of claim 8, wherein the change in the gradient of the magneticfield is a higher-than-second order gradient.
 11. The method of claim 8,further comprising mechanically coupling a third magnet to the secondmagnet on a side opposite the first magnet to move together with thesecond magnet along the polarization axis, wherein: the third magnet ispositioned such that a pair of like magnetic poles of the third magnetand the second magnet face in opposite directions; and the change in thegradient of the magnetic field is a third order gradient.
 12. The methodof claim 8, wherein the first magnet and the second magnet arepositioned with a separation distance of substantially 2 mmtherebetween.
 13. The method of claim 8, wherein the sensing system is amicroscope, measuring the change in the gradient of the magnetic fieldincludes determining, with the microscope, a change in deflection of thefirst magnet and/or the second magnet along the polarization axis,wherein the change in the gradient of the magnetic field is determinedbased on the change in deflection.
 14. A method of assembling a magnetarray for a gradiometer comprising: attaching a first magnet to a topsurface of a first transparent substrate, the first transparentsubstrate having a greater width than the first magnet; attaching asecond magnet to a top surface of a second transparent substrate, thesecond transparent substrate having a greater width than the secondmagnet; positioning the second magnet over the first magnet along adipole axis; aligning the first magnet and the second magnet along thedipole axis by viewing the first magnet and second magnet with amicroscope from a position directly above the second magnet along thedipole axis and adjusting the position of the first magnet and/or thesecond magnet; placing epoxy on top of the first magnet; and loweringthe second magnet onto the first magnet while continuing to view thefirst magnet and the second magnet with the microscope to maintainalignment along the dipole axis.
 15. The method of claim 14, wherein:the epoxy is UV curable epoxy which is cured with UV radiation to attachthe first magnet and the second magnet; the first magnet is held to thefirst transparent substrate with cured UV epoxy; and the second magnetis held to the second transparent substrate with cured UV epoxy.
 16. Themethod of claim 14, wherein the first transparent substrate is a glasscoverslip and the second transparent substrate is a glass coverslip. 17.The method of claim 16, further comprising, after the step of loweringthe second magnet onto the first magnet: curing the epoxy to attach thefirst magnet to the second magnet; breaking and removing the firsttransparent substrate from the first magnet; and breaking and removingthe second transparent substrate from the second magnet.
 18. The methodof claim 14, wherein the second magnet is attached to the secondtransparent substrate with a like polar orientation to the first magnet.19. The method of claim 14, wherein the first magnet is substantiallythe same size as the second magnet.